2 432 CHAPTER 6 POLYNOMIALS AND POLYNOMIAL FUNCTIONS The first step in learning to factor a trinomial is to identify its coefficients. To be consistent, we first write the trinomial in standard ax 2 bx c form, then label the three coefficients as a, b, and c. Example 2 Identifying the Coefficients of ax 2 bx c When necessary, rewrite the trinomial in ax 2 bx c form. Then label a, b, and c. (a) x 2 3x 18 a 1 b 3 c 18 x 2 24x 23 a 1 b 24 c 23 (c) x x First rewrite the trinomial in descending order. x 2 11x 8 Then, a 1 b 11 c 8 CHECK YOURSELF 2 When necessary, rewrite the trinomial in ax 2 bx c form. Then label a, b, and c. (a) x 2 5x 14 x 2 18x 17 (c) x 6 2x 2 Not all trinomials can be factored.to discover if a trinomial is factorable, we try the ac test. Rules and Properties: The ac Test A trinomial of the form ax 2 bx c is factorable if (and only if) there are two integers, m and n, such that ac mn and b m n In Example 3, we will determine whether each trinomial is factorable by finding the values of m and n. Example 3 Using the ac Test Use the ac test to determine which of the following trinomials can be factored. Find the values of m and n for each trinomial that can be factored. (a) x 2 3x 18 First, we note that a 1, b 3, and c 18, so ac 1( 18) 18. Then, we look for two numbers, m and n, such that mn ac and m n b. In this case, that means mn 18 m n 3

4 434 CHAPTER 6 POLYNOMIALS AND POLYNOMIAL FUNCTIONS There is no need to go further. We have found two integers with a product of 30 and a sum of 7. So m 10 and n 3. In this example, you may have noticed patterns and shortcuts that make it easier to find m and n. By all means, use those patterns. This is essential in mathematical thinking. You are taught a step-by-step process that will always work for solving a problem; this process is called an algorithm. It is very easy to teach a computer an algorithm. It is very difficult (some would say impossible) for a computer to have insight. Shortcuts that you discover are insights. They may be the most important part of your mathematical education. CHECK YOURSELF 3 Use the ac test to determine which of the following trinomials can be factored. Find the values of m and n for each trinomial that can be factored. (a) x 2 7x 12 x 2 5x 14 (c) 2x 2 x 6 (d) 3x 2 6x 7 So far we have used the results of the ac test only to determine whether a trinomial is factorable. The results can also be used to help factor the trinomial. Example 4 Using the Results of the ac Test to Factor a Trinomial Rewrite the middle term as the sum of two terms, then factor by grouping. (a) x 2 3x 18 We see that a 1, b 3, and c 18, so ac 18 b 3 We are looking for two numbers, m and n, so that mn 18 m n 3 In Example 3, we found that the two integers were 3 and 6 because 3( 6) 18 and 3 ( 6) 3. That result is used to rewrite the middle term (here 3x) as the sum of two terms. We now rewrite the middle term as the sum of 3x and 6x. x 2 3x 6x 18 Then, we factor by grouping: x 2 3x 6x 18 x(x 3) 6(x 3) (x 3)(x 6) x 2 24x 23 We use the results of Example 3, in which we found m 1 and n 23, to rewrite the middle term of the expression. x 2 24x 23 x 2 x 23x 23 Then, we factor by grouping: x 2 x 23x 23 x(x 1) 23(x 1) (x 1)(x 23)

5 FACTORING TRINOMIALS: THE ac METHOD SECTION (c) 2x 2 7x 15 From example 3(d ), we know that this trinomial is factorable and that m 10 and n 3. We use that result to rewrite the middle term of the trinomial. 2x 2 7x 15 2x 2 10x 3x 15 2x(x 5) 3(x 5) (x 5)(2x 3) CHECK YOURSELF 4 Rewrite the middle term as the sum of two terms, then factor by grouping. (a) x 2 7x 12 x 2 5x 14 (c) 2x 2 x 6 (d) 3x 2 7x 6 Not all product pairs need to be tried to find m and n. A look at the sign pattern will eliminate many of the possibilities. Assuming the lead coefficient to be positive, there are four possible sign patterns. Pattern Example Conclusion 1. b and c are both positive. 2x 2 13x 15 m and n must be positive. 2. b is negative and c is x 2 3x 2 m and n must both be negative. positive. 3. b is positive and c is x 2 5x 14 m and n are of opposite signs. negative. (The value with the larger absolute value is positive.) 4. b and c are both negative. x 2 4x 4 m and n are of opposite signs. (The value with the larger absolute value is negative.) Sometimes the factors of a trinomial seem obvious. At other times you might be certain that there are only a couple of possible sets of factors for a trinomial. It is perfectly acceptable to check these proposed factors to see if they work. If you find the factors in this manner, we say that you have used the trial and error method. This method is discussed in Section 6.7*, which follows this section. To this point we have been factoring polynomial expressions. When a function is defined by a polynomial expression, we can factor that expression without affecting any of the ordered pairs associated with the function. Factoring the expression makes it easier to find some of the ordered pairs. In particular, we will be looking for values of x that cause f(x) to be 0. We do this by using the zero product rule. Rules and Properties: Zero Product Rule If 0 ab, then either a 0, b 0, or both are zero. Another way to say this is, if the product of two numbers is zero, then at least one of those numbers must be zero.

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